Welcome to your ACT 1 Geo: Recognize Pythagorean relationships

$Q_{1}:$ The cube in figure below has edges of length $5$. What is the distance from vertex $H$ to vertex $K$?

$Q_{2}:$ If the sides of a right triangle have lengths $x − 3$, $x + 1$, and $x + 5$, then $x =$?

$Q_{3}:$ In the figure below, square $CDEF$ shares the common side $CF$ with parallelogram $ABCF$. The area of the square is $225$, the length of $AG$ is $9$, and the length of $FG$ is $21$. What is the area of the parallelogram?

$Q_{4}:$ Triangle $ABC$ has a right angle at vertex $A$. If $sin C = \frac{5}{13}$, what is the value of $tan B$?

$Q_{5}:$ As shown in the figure below, a lifeguard sees a struggling swimmer who is $40$ feet from the beach. The lifeguard runs $60$ feet along the edge of the water at a speed of $12$ feet per second. She pauses for $1$ second to locate the swimmer again and then dives into the water and swims along a diagonal path to the swimmer at a speed of $5$ feet per second. How many seconds go by between the time the lifeguard sees the struggling swimmer and the time she reaches the swimmer?

$Q_{6}:$ Find the length of a rectangle whose width is $5 $cm and whose diagonal measures $13 $cm.

$Q_{7}:$ Valerie drives $10 $ miles due east, then drives $20$ miles due north, and finally drives $5$ miles due west. Which of the following represents the straight-line distance Valerie is from her starting point?

$Q_{8}:$ In figure below, what is the value of $d$?

$Q_{9}:$ What is the area of $\Delta LPN $?

$Q_{10}:$ In the rectangular solid shown, if $AB = 4$, $BC = 3$, and $BF = 12$, what is the perimeter of triangle $EDB $?

$Q_{11}:$ In the cube below, $M$ is the midpoint of $BC$, and $N$ is the midpoint of $GH$. If the cube has a volume of $1$, what is the length of $MN $?

$Q_{12}:$ What is the distance between the origin and the point $(5, 6, 7) $?

$Q_{13}:$ Sphere $O$ has a radius of $6$, and its center is at the origin. Which of the following points is NOT inside the sphere?

$Q_{14}:$ What is the distance between the $x$-intercept and the $y$-intercept of the line given by the equation $2y = 6 − x $?

$Q_{15}:$ Line segments $AC$ and $BD$ intersect at point $O$, such that each segment is the perpendicular bisector of the other. If $AC = 7$ and $BD = 6$, then $sin \angle ADO =$

$Q_{16}:$ A rectangular room has walls facing due north, south, east, and west. On the southern wall, a tack is located $85$ inches from the floor and $38$ inches from the western wall, and a nail is located $48$ inches from the floor and $54$ inches from the western wall. What is the distance in inches between the tack and the nail?

$Q_{17}:$ In figure below, $AB = 4$, $BC = 7$, and $CD = 1$. If $AC$ is a diameter of the circle, then what is the length of $AD $?

$Q_{18}:$ Find $sin \angle BAC =$

$Q_{19}:$ In the following figure, the ratio of $x$ to $y$ is $1:4$. What is the value of $x$?

$Q_{20}:$ In the following figure, $ABCD$ is a rectangle, and $PQR$ is an equilateral triangle. Given the provided measurements are in inches, what is the area of the rectangle $ABCD$ in square inches?

$Q_{21}:$ The Great Pyramid of Giza was constructed around $2560$ BC in Egypt and is the only one of the Seven Wonders of the Ancient World that is still intact. As seen in the following diagram, the pyramid has a square base with side lengths of $230$ meters, and it has a vertical height of $147$ meters. The line marked “s” intersects the base of the pyramid at a right angle. What is the length of line “s” rounded to the nearest meter?

$Q_{22}:$ In the following figure, the area of the circle centered at the point $O$ is $\frac{29 \pi}{4}$ square feet. Given that $AB$ has a length of $2$ feet, what is the area of triangle $ABC$ in square feet?

$Q_{23}:$ The length of one leg of a right triangle is three times as large as the length of the other leg. If the hypotenuse has a length of $10$ inches, which of the following is the perimeter of the triangle in inches?

$Q_{24}:$ In square centimeters, what is the area of the parallelogram in the following figure?

$Q_{25}:$ An explorer hikes North from her camp for eight miles and then hikes East for six miles. Assuming the explorer takes the most direct route back to her starting point, how far is she from her camp?